Problem: If we express $3x^2 - 6x - 2$ in the form $a(x - h)^2 + k$, then what is $a + h + k$?
Solution: We complete the square.  First, we factor 3 out of the terms $3x^2 - 6x$ to get $3(x^2 - 2x)$.  We can square $x - 1$ to get $x^2 - 2x + 1$, so $3(x^2 - 2x) = 3[(x - 1)^2 - 1] = 3(x - 1)^2 - 3$, and \[3(x^2 - 2x) - 2 = 3(x - 1)^2 - 3 - 2 = 3(x - 1)^2 - 5.\]We see that $a = 3$, $h = 1$, and $k = -5$, so $a + h + k = 3 + 1 + (-5) = \boxed{-1}$.